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In order to evaluate the stationary point (the point in which the phase is minimum or maximum),
we need to solve the equation

| |
(25) |

Equation (B-1) reduces to the condition that the horizontal projection for rays from each of
the source and receiver to the specular reflection point (SRP) add up to equal the source-receiver
offset, *X* (Popovici, 1995).
Solving equation (B-1) in its full form involves solving a
sixth-degree polynomial, which I prefer to avoid doing
analytically.
Setting *p*_{x}=0 in equation (B-1) yields solutions (stationary points) for reflections from
horizontal reflectors,
and, therefore, equation (B-1) reduces to

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(26) |

with a stationary point given by
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(27) |

In fact, *p*_{h0} is the exact stationary point solution for horizontal reflections,
which can be derived directly from the hyperbolic moveout equation:
Where ,
which is the same as equation (B-3).
Expanding equation (B-1) using Taylor's series around *p*_{h}=0,
and ignoring terms beyond the linear in *p*_{h}, yields

with a stationary point given by
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(28) |

Equation (B-4) corresponds to the exact solution for a vertical reflector,
where *p*_{h}=0 and . It also provides a good approximation away
from . Figure B-1 shows a comparison between the exact *p*_{h} solution of
equation (B-1) (obtained numerically)
and that given by equation (B-4) as a function of *p*_{x} for three sets of . The
absolute difference between the two solutions is also displayed. As expected, errors increase with
offset, and
**ph1eta0m
**

Figure 17 Left: Values of *p*_{h} as a function *p*_{x} calculated numerically
(solid curves), and calculated analytically (dashed curves)using equation (B-4).
Right: The absolute difference
between the two curves on the left. The medium is homogeneous and isotropic with *v*=2.0 km/s.
The black curve corresponds to =1.0 km/s, the dark-gray curve corresponds to
=2.0 km/s, and the light gray curve corresponds to =3.0 km/s.

the accuracy
of the approximate solution reduces at *p*_{x}=0. In fact, for *p*_{x}=0, *p*_{h} in
equation (B-4)
equals which is clearly different from the exact solution given by
equation (B-3). Inserting the exact solution, in place of the factor
, in equation (B-4) yields a new approximate solution for *p*_{h} given by

| |
(29) |

This solution is exact for *p*_{x}=0 and , and therefore, I will refer
to it as the 2-point solution (it fits exactly at two points).
Figure B-2 shows a comparison
between the exact solution of equation (B-1) (obtained numerically)
and that given by equation (B-5), again for three sets of .Clearly, this new curve of *p*_{h} better fits the exact solution than that given in Figure B-1.
Although
the percentage of error in *p*_{h} seems to be high, the corresponding error in the phase estimate
(which is the key quantity used in the migration) is much lower
because , as depicted in Figure 1,
is rather insensitive to *p*_{h}, especially around the exact solution.
In other words, the curvature of the phase function around its minimum is low. Such low curvature, according
to equation (A-2), will result in a greater contribution in terms of amplitude.
This translates to practically
no error when it comes to actual geophysical applications.
**ph2eta0m
**

Figure 18 Same as in Figure B-1, but with the analytical solution evaluated
using the 2-point fitting of equation (B-5).

Let's find yet another exact solution,
that is, the solution when *p*_{s}=0 (*p*_{x}=*p*_{h}). The reflector angular
correspondence of this approximation depends on the offset. In this case,
equation (B-1) reduces to

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(30) |

with a solution for *p*_{h} given by
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(31) |

To insure that *p*_{h} given by equation (B-5) has the value of *p*_{hs}
as *p*_{x}=*p*_{h}, we must insert, some how, equation (B-7)
into equation (B-5). This task is accomplished by forcing *p*_{h},
given by equation (B-5), to linearly equal
*p*_{hs} at *p*_{x}=*p*_{h},
and, thus, equation (B-5) becomes
| |
(32) |

The term between the square brackets is to insure this linear fit.
To solve for *a* and *b* in equation (B-8), *p*_{h} must satisfy two conditions:
(1) *p*_{h} should reduce to the form given by equation (B-3) for *p*_{x}=0, and this condition
will result in *b*=1.
(2) *p*_{h} should reduce to the form given by equation (B-7) for *p*_{x}=*p*_{hs}.
The second condition, after some mathematical manipulation, yields
Therefore, *p*_{h} given by equation (B-8) is an exact solution of equation (B-1)
at three points (*p*_{x}=0, *p*_{x}=*p*_{hs}, and ) and a good approximation, as
demonstrated by Figure B-3, elsewhere. I will refer to this equation as the 3-point
solution.
Figure B-3 shows *p*_{h} given by equation (B-8) next to the exact solution for *p*_{h}.
Clearly, the three point fitting given by equation (B-8) resulted in a good approximation
of *p*_{h}. The accuracy of this approximation will be better appreciated when we observe the low amount of
errors in traveltime calculation induced by this approximation (i.e., Figure 3).

**ph3eta0m
**

Figure 19 Same as in Figure B-1, but with the analytical solution evaluated
using the 3-point fitting of equation (B-8).

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** Up:** Analytical Approximations of the
** Previous:** Analytical Approximations of the
Stanford Exploration Project

11/11/1997